Day 3: Missing Data in Longitudinal and Multilevel Models

by Levente (Levi) LittvayCentral European UniversityDepartment of Political Sciecelevente@littvay.hu

Multilevel and Longitudinal Models• Longitudinal SEM (Latent Growth Curve)

– Structural Equation Models– Most approaches that work with SEMs work– There are model size and identification issues– (Traditionally use) Direct Estimation

• Multilevel / Mixed / Random Effect Models– Pattern problems– Level problems– What to model and what not to model issues– (Traditionally use) Imputation

Missing Data in Longitudinal Structural Equation Models

Missing Data in SEMs

• Same approaches work• Direct Estimation

– More Common Approach– Missing can only be on the DV

(usually not an issue with longitudinal models)• Imputation

– Can impute with an unstructured model– AMOS can impute using the analysis model

(If no missing on the exogenous variables)

Longitudinal SEM

• Example - Latent Growth Curve• It is just a structural equation model• All observed variables are DVs

from Mplus Manual (ex 6.1)

Auxiliary Variables• Just include them as you would otherwise

– MI: include them in the imputation model– Direct estimation: correlate them with each

other and all other observed variables• Practical Issues

– Can get out of hand• Imputation: Convergence + Model Size• Direct Estimation: Model Size + Convergence

– Identification issues correlation of ~1 is not a unique information in the correlation matrix

– Could collapse (if it still informs missingness)

Planned Missing• Rolling Panel

– You return to each person twice– You measure over a longer period of time– Can reduce panel effect

– Always test power and convergence

Attrition

• If attrition is MAR you are fine– Ask questions like how likely are you to

come back next time. etc.

• If not NMAR you are not fine

Extension of the Heckman Model

• The analytical model is estimated simultaneously with the model of missingness

• Mplus Mailing List (Moh-Yin Chang - SRAM)• Model Dropout (with a Survival Model)

simultaneously with the Longitudinal Model• Let Residuals Correlate• Pray that it Runs

Multilevel Models

Stacked Dataset Patterns

Example (My Dissertation)

• Over time data on 186 countries (1984-2004)• Item Missing (Hungary Trade Volume 1991)• A variable missing for a whole country

(Had corruption data for 143 countries.)• No data at all on Afghanistan, Cuba and

North Korea (Unit Missing?)• No data on energy consumption for 2004• No data on West Germany after 1989

(Should that even be treated as missing?)

MLM Missing Data

• You are OK with MAR missing on the DV

• You are OK with MAR wave missing– But if you have any information on the wave

it will not be incorporated in the model– It is better to incorporate all info to help

satisfy the MAR assumption

Multiple Imputation for Multilevel Models

MLM Imputation Procedures

• OK for Level 1 Missing Data– PAN (Schafer, Bayesian, S-Plus/R module)– MlWin (Implemented Schafer’s PAN - Better)– WinMICE (Chained Equations)– Amelia II (Not true multilevel model)

• Upcoming: Shrimp (Yucel)

Imputation Model (Level 1)

• Thinking about the missing data model for multilevel models. (Conceptually Difficult)– Conventional Wisdom: Missing data model

should be the same as the analysis model plus auxiliary variables.

– Unstructured Model• Issues

– Inclusion of random effects for aux variables– Centering– Interactions

Bayesian Convergence

• Markov Chain Monte Carlo• Random Walk Simulation• Problem of autoregressive behavior• Independent random draws produce the

“posterior distribution” that imputations are sampled from.

• Bayesian convergence is in the eye of the beholder. No standard rules.

Ocular Shock Test of Convergence

• Well Implemented in MI software• Has to be evaluated for all estimated

parameters (this really sucks)• Two Plots to Assess:

– Parameter Value Plot– Autocorrelation Function Plot

• Be careful about the range of assessment• Worst linear function - lucky if available

Quickly Converging Model

Slowly Converging Model

Pathological SituationNo Convergence

Did Not Yet Reach Convergence

Pseudo Multilevel Model• Random Effect of the Intercept

– Dummies for each level 1 unit (but one)– Pro: no distributional assumption of the

variance of the intercept– Con: eats up degrees of freedom

• Random Effects of slopes– Interaction between the above dummy and

the independent variable– Same pros and cons

• Same can be done with imputation model– Impact of ignoring random effects?

Level 2 missing (sucks)• If you do Schafer suggests the following

– Collapse your level 1 variables by averaging across your level 2 unitsThis produces a single level dataset

– Impute the single level dataset 10 times(Use a single level procedure)

– Take the 10 level 2 datasets remerge them with the level 1 data (exclude?)

– Impute level 1 missing once for each 10 using a multilevel imputation technique

• Assumptions of this approach (iterative?)

MI Support in Software

• HLM and Mplus• Maybe Stata (clarify, micombine - ?,?)• Maybe R (zelig - ?)• MlWin can do imputation

May also combine (possibly with hacking)

Rubin’s Rules

• Combining results is still easy• Use NORM like for single dataset• One point of confusion is random effects• But they also have parameter estimates

and standard errors• Combine like you combine coefficients

and standard errors• Don’t forget about the error covariances

Direct Estimation of Multilevel Models

Direct Estimation of MLMs

• It is computationally intensive(requires numerical integration)

• Level 1 missing seems OK• Missing IVs: make IVs into DVs• Problem of auxiliary variables

Implementation

• In Mplus– Same as with SEM models– Multilevel SEM model– Downside: limited to unstructured error

covariance matrix. (No AR1 band-diagonal)• Mplus does level 2 missing with

monte-carlo integration– Unstable

• MlWin’s multilevel factor analysis (??)

Practical Considerations

• Getting good starting values– Really easy for most models– Run the model with all complete cases– Take results and use as starting values– Tedious, but worth it